zero divisor graph‎

Let $\mathcal G = (\mathcal V, \mathcal E)$ be a simple graph, an $L(2,1)$-labeling of $\mathcal G$ is an assignment of labels from non-negative integers to vertices of $\mathcal G$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The $\lambda$-number of $\mathcal G$, denoted by $\lambda(\mathcal G)$, is the smallest positive integer $\ell$ such that $\mathcal G$ has an $L(2,1)$-labeling with all labels as  members of the set $\{ 0, 1, \dots, \ell \}$. The zero-divisor graph of a finite commutative ring $R$ with unity, denoted by $\Gamma(R)$, is the simple graph whose vertices are all zero divisors of $R$ in which two vertices $u$ and $v$ are adjacent  if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labeling of some  zero-divisor graphs. We study the \textit{partite truncation}, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between  $\lambda$-numbers of the graph  and its partite truncated one. We make use of the operation \textit{partite truncation} to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.

In this article, we discussed the zero-divisor graph of a commutative ring with identity $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$ where $u^3=0$ and $p$ is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zero-divisor graph $\Gamma(R).$ Also, we find the eigenvalues, energy and spectral radius  of both adjacency and Laplacian matrices of $\Gamma(R).$

Let $R$ be a finite commutative ring with or without unity and $\Gamma_{e}(R)$ be its extended zero-divisor graph with vertex set $Z^{*}(R)=Z(R)\setminus \lbrace0\rbrace$ and two distinct vertices $x,y$ are adjacent if and only if $x.y=0$ or $x+y\in Z^{*}(R)$. In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number $1 ~ \text{or}~ 2$. We completely characterize the rings of the form $R\cong R_1\times R_2 $, where $R_1$ and $R_2$ are local, having clique number $3,~4~\text{or}~5$. Further we determine the rings of the form $R\cong R_1\times R_2 \times R_3$, where $R_1$,$R_2$ and $R_3$ are local rings, to have clique number equal to six.

The graph Γ(R1◦R2) of the lexicographic product of two commutative rings R1; R2 is considered. It was shown that Γ(R1 ◦ R2) is connectedand diam(Γ(R1 ◦ R2)) ≤ 2. We get the several expressions for finding theconnectivity κ(Γ(R1 ◦ R2)) when certain conditions are given

Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The annihilator intersection graph of $R$, represented by $AIG(R)$, is an undirected graph with $A(R)^*$ as the vertex set and $\mathfrak{M} \sim \mathfrak{N}$ is an edge of $AIG(R)$ if and only if $Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N})$, for distinct vertices $\mathfrak{M}$ and $\mathfrak{N}$ of $AIG(R)$. In this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.

Let $R$ be a commutative ring and $\Gamma(R)$  be its zero-divisor graph. All the vertices of zero divisor graphs are the non-zero divisors of the commutative ring, with two distinct vertices joined by an edge in case their product in the commutative ring is zero. In this paper, we study the metric dimension and neighbourhood resoling set for the zero divisor graphs of order 3,4,5,6,7,8,9,10 of a small finite commutative ring with a unit.

‎This paper introduces the concepts of reproduced general hyperring and valued-orderable general hyperring and investigates some properties of these classes of general hyperrings‎. ‎It presents the notions of‎ ‎zero divisors and zero  divisor graphs are founded on the absorbing elements of general hyperrings‎. ‎General hyperrings can have more than one zeroing element‎, ‎and therefore‎, ‎based on the zeroing elements‎, ‎multiple zero divisors can be obtained‎. ‎In this study‎, ‎we  discuss the isomorphism of zero divisor graphs based on the diversity of divisors of zero divisors‎. ‎The non-empty  intersection of the set of absorbing elements and the hyperproduct of zero divisors of general hyperrings play a major  role in the production of zero divisor graphs‎. ‎Indeed it investigated a type classification of zero divisor graphs based on  the finite general hyperrings‎. ‎We discuss the finite reproduced general hyperrings‎, ‎investigate their zero divisor graphs‎, ‎and show that an infinite reproduced general hyperring can have a finite zero divisor graph‎.

For a commutative ring $R$ with identity $1neq 0$, let the set $Z(R)$ denote the set of zero-ivisors and let $Z^{*}(R)=Z(R)setminus {0}$ be the set of non-zero zero-divisors of $R$.  The zero-divisor graph of $R$, denoted by $Gamma(R)$, is a simple graph whose vertex set is $Z^{*} (R)$ and two vertices $u, v in Z^*(R)$ are adjacent if and only if $uv=vu=0$. In this article, we find the signless Laplacian spectrum of the zero divisor graphs $ Gamma(mathbb{Z}_{n}) $ for $ n=p^{M_{1}}q^{M_{2}}$, where $ p<q $ are primes and $ M_{1} , M_{2} $ are positive integers.

For a commutative ring $R$ with identity $1\neq 0$, let $Z^{*}(R)=Z(R)\setminus \lbrace 0\rbrace$ be the set of non-zero zero-divisors of $R$, where $Z(R)$ is the set of all zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*}(R)=Z(R)\setminus \{0\}$ and two vertices of $ Z^*(R)$ are adjacent if and only if their product is $ 0 $. In this article, we find the structure of the zero-divisor graphs $ \Gamma(\mathbb{Z}_{n}) $, for $n=p^{N_1}q^{N_2}r$, where $2<p<q<r$ are primes and $N_1$ and $N_2$ are positive integers.

In this paper, we investigate some of the graph energies of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$. Let $Z(R)$ be the set of zero-divisors of a commutative ring $R$ with non-zero identity and $Z^*(R)=Z(R)\setminus \{0\}$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set in $Z^*(R)$ and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$. We investigate some energies of $\Gamma(R)$ for the commutative rings $R\simeq \mathbb{Z}_{p^2}\times \mathbb{Z}_{q}$, $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ and $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ where $p, q$ the prime numbers.

In this paper, some graph parameters of the zero-divisor graph $\Gamma(R)$ of a finite commutative ring $R$ for $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{2p}$ where $p>2$ a prime, are investigated. The graph $\Gamma(R)$ is a simple graph whose vertex set is the set of non-zero zero-divisors of a commutative ring $R$ with non-zero identity and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$.
In this paper, we study some of the topological indices such as graph energies, the Zagreb indices and the domination parameters of graphs $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{p^2} \big)$ and $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{2p}\big)$.

For a finite commutative ring $ mathbb{Z}_{n} $ with identity $ 1neq 0 $, the zero divisor graph $ Gamma(mathbb{Z}_{n}) $ is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices $ x $ and $ y $ are adjacent if and only if $ xy=0 $. We find the Randi'c spectrum of the zero divisor graphs $ Gamma(mathbb{Z}_{n}) $, for various values of $ n$ and characterize $ n $ for which $ Gamma(mathbb{Z}_{n}) $ is Randi'c integral.

Let $R$ be a commutative ring with unity not equal to zero and let $Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) subseteq V(G)$ such that $|N(v) cap C(G)| = 1$ for all $v in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. We also show that if $Gamma(R)$ is a regular graph on $|Z^*(R)|$ number of vertices, then $R$ is a reduced ring and $|Z^*(R)| equiv 0 (mod ~2)$, where $Z^*(R)$ is a set of non-zero zero-divisors of $R$. We provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. Finally, we determine the cardinality of a total perfect code in $Gamma(R)$ and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity.

* Formulas are not displayed correctly.

 

Let M(X,A,μ) be the ring of real-valued measurable functions on a measurable space (X,A) with measure μ. In this paper, we study the zero-divisor graph of M(X,A,μ), denoted by Γ(M(X,A,μ)). We give the relationships among graph properties of Γ(M(X,A,μ)), ring properties of M(X,A,μ) and measure properties of (X,A,μ). Finally, we investigate the continuity properties of Γ(M(X,A,μ)).

We study Beck-like coloring of measurable functions on a measure space Ω taking values in a measurable semigroup ∆. To any measure space Ω and any measurable semigroup ∆, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Ω and having values in ∆, with two vertices f and g adjacent if f · g = 0 a.e.. We show that, if Ω is atomic, then not only the Beck’s conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph.

In this work, the radius, diameter and a chromatic number of zero divisor graph of the ring Zn for some n are been determined. These graphs are Γ(Zp2q2), Γ(Zp2), Γ(Zpq), Γ(Zp3), Γ(Zp2q) and Γ(Zpqr). Furthermore, the largest induced subgraph isomorphic to complete subgraph in the graph Γ(Zp3) and Γ(p2q) are calculated.

‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎The‎ ‎annihilator graph of $M$‎, ‎denoted by $AG(M)$ is a simple undirected‎ ‎graph associated to $M$ whose the set of vertices is‎ ‎$Z_R(M) setminus {rm Ann}_R(M)$ and two distinct vertices $x$ and‎ ‎$y$ are adjacent if and only if ${rm Ann}_M(xy)neq {rm‎ ‎Ann}_M(x) cup {rm Ann}_M(y)$‎. ‎In this paper‎, ‎we study the‎ ‎diameter and the girth of $AG(M)$ and we characterize all modules‎ ‎whose annihilator graph is complete‎. ‎Furthermore‎, ‎we look for the‎ ‎relationship between the annihilator graph of $M$ and its zero-divisor‎ ‎graph‎.

‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this paper is to study the interplay of the properties of submodule $N$ and‎ ‎the properties of $Gamma(M‎, ‎N)$‎.

Let R be an associative ring with identity. A ring R is called reversible if ab=0, then ba=0 for a,b∈R. The quasi-zero-divisor graph of R, denoted by Γ∗(R) is an undirected graph with all nonzero zero-divisors of R as vertex set and two distinct vertices x and y are adjacent if and only if there exists 0≠r∈R∖(ann(x)∪ann(y)) such that xry=0 or yrx=0. In this paper, we determine the diameter and girth of Γ∗(R). We show that the zero-divisor graph of R denoted by Γ(R), is an induced subgraph of Γ∗(R). Also, we investigate when Γ∗(R) is identical to Γ(R). Moreover, for a reversible ring R, we study the diameter and girth of Γ∗(R[x]) and we investigate when Γ∗(R[x]) is identical to Γ(R[x]).